The phonon thermal Hall angle in black phosphorus

The origin of phonon thermal Hall Effect (THE) observed in a variety of insulators is yet to be identified. Here, we report on the observation of a thermal Hall conductivity in a non-magnetic elemental insulator, with an amplitude exceeding what has been previously observed. In black phosphorus (BP), the longitudinal (κii), and the transverse, κij, thermal conductivities peak at the same temperature and at this peak temperature, the κij/κjj/B is ≈ 10−4−10−3 T−1. Both these features are shared by other insulators displaying THE, despite an absolute amplitude spreading over three orders of magnitude. The absence of correlation between the thermal Hall angle and the phonon mean-free-path imposes a severe constraint for theoretical scenarios of THE. We show that in BP a longitudinal and a transverse acoustic phonon mode anti-cross, facilitating wave-like transport across modes. The anisotropic charge distribution surrounding atomic bonds can pave the way for coupling between phonons and the magnetic field.

Thermal Hall effect (THE) refers to the emergence of a transverse thermal current by a longitudinal thermal gradient, in presence of a magnetic field. In metals, it is intimately linked to the electrical Hall effect through the Wiedemann-Franz law. Following its original discovery 1 , thermal Hall effect was observed in a wide variety of insulating solids [2][3][4][5][6][7][8][9][10] . Its origin is controversial in exotic cases [11][12][13][14] , but in strontium titanate 7 , a non-magnetic insulator, THE is undoubtedly caused by phonons and is drastically reduced by the introduction of extrinsic atoms 15,16 .
What are the minimal ingredients required to produce a detectable THE? To what extent the effect seen in various families share a common origin? Here, we address these two questions by reporting on the observation of thermal Hall effect in black phosphorus (BP), the simplest insulator known to display THE. It is not only non-magnetic, but also lacking ionic bonds. Interestingly, while the amplitude of the thermal Hall conductivity in BP exceeds what was found in all other insulators, the transverse and the longitudinal thermal conductivities peak at the same temperature and their ratio has a similar amplitude. We note that the charge distribution is anisotropic and atomic vibrations can therefore respond to the magnetic field. We show that two out of the three acoustic branches anti-cross and are close to degeneracy in the momentum space, and therefore, energy transfer across harmonic vibrational states 26,27 plays a role in setting the amplitude of thermal conductivity and since the charge distribution is anisotropic, magnetic field can couple to atomic vibrations. The austere context of our experimental observation strongly constrains theoretical scenarios. Figure 1c displays the temperature dependence of the longitudinal thermal conductivity. As found in previous studies 31,32 , there is a large anisotropy. κ xx peaks to 311 WK −1 m −1 at 27 K and κ zz to 1770 WK −1 m −1 at 34 K. This is a consequence of the larger sound velocity along the zigzag direction.
The field dependent thermal Hall angle ∇ T j / ∇ T i , is shown in Fig. 1d, e for two different configurations and at four different temperatures. In both configurations, it is linear in magnetic field. Interestingly, ∇ T j / ∇ T i is anisotropic: It attains 5.0 × 10 −3 at 12 T and 35.3 K for one configuration and 1.2 × 10 −3 at 12 T and 32.1 K for the other. This fourfold anisotropy is clearly seen in the temperature dependence of ∇ T j /∇ T i (Fig. 1f).
Combining the thermal Hall angle (Fig. 1f), the longitudinal thermal conductivity (Fig. 1c), leads us to the thermal Hall conductivity, shown in Fig. 2a. The inset shows the contribution of electrons to THE can be estimated through the Wiedemann-Franz law. At low temperatures, κ e ij ≈ L 0 σ ij T ≪ κ ij and therefore electrons can be totally neglected below 50 K and near the peaks in thermal conductivities. Strikingly, κ zx and −κ xz become very close to each other. Within experimental margin, they are equal, as expected by the Onsager reciprocal relations for diffusive transport (κ ij (H) = κ ji ( − H) = − κ ji (H)) 1,33 . The inequality between the two off-diagonal components at high temperature can be tracked to the gradual emergence of a sizeable thermoelectric response at high temperature and therefore, a significant difference between the measured thermal conductivity (in absence of charge current) and the true Onsager coefficient (which is thermal conductivity in absence of the electric field). A similar phenomenon was observed in the case of dilute metallic strontium titanate 16 (see the Supplementary Material for more details). Figure 2b, c compare the temperature dependence of longitudinal and transverse thermal conductivity. Multiplying κ xz (T) and κ zx (T) by a factor of −150 and 1000 respectively, one finds that κ ij (T) and κ ii (T) peak almost at the same temperature for both configurations. It's worth noting that the κ ij /κ ii ratio is anisotropic. This is unavoidable and due to the combination of the Onsager reciprocity (κ ij = −κ ji ) and the anisotropy of longitudinal thermal conductivity (κ ii ≠ κ jj ).

Universal features of thermal Hall effect in different insulators
Let us now compare BP with other insulators. The peak amplitude of κ ij in BP~2.2 WK −1 m −1 is four orders of magnitude larger than what was seen in Tb 3 Ga 3 O 12 1 , almost two orders of magnitude larger than the values reported for cuprates 5,6 and SrTiO 3 7 , and almost twice what was recently reported in Cu 3 TeO 6 9 . On the other hand, the thermal Hall angle remains in the same narrow range of 0.1-1% under a magnetic field of about 10 T. As seen in Fig. 2d, across several orders of magnitude variation, longitudinal and transverse thermal conductivity scale with each other. Using, the magnetic length, ' B = ffiffiffiffiffiffiffiffiffiffi _=eB p one can extract a length scale, λ tha , from this angle: In the case of electrons in the weak-field limit, extracting a length in this way from the Hall angle will give rise to a geometric average of the mean-freepath and the Fermi wavelength. In a similar manner, in some scenarios for phonon Hall response, one expects a correlation between the Hall angle and the mean-free-path.
Intriguingly, for all these solids, λ tha remains between 2 and 7 angstroms. Such a length is comparable to the shortest possible phonon wavelength allowed by the interatomic distance. Our observation that it does not correlate with the phonon mean-free-path, which varies by more than three orders of magnitude among these solids, excludes many (but not all) 'extrinsic' scenarios.
A second universal feature is shown in Fig. 3. As first noted in ref. 7 , the transverse and longitudinal conductivities occur peak almost the same temperature in all insulators displaying THE. Thus, the thermal Hall response is always maximal when the wave-vector of the heatcarrying acoustic phonons have sufficiently shrunk to make Umklapp scattering irrelevant, but boundary scattering is not yet the dominant scattering mechanisms.  c Thermal conductivity along the armchair (κ xx ) and the zigzag (κ zz ) orientations.
There is a large in-plane anisotropy, as found by previous studies 31,32 . d, e Field dependence of the thermal Hall angle at different temperatures with the heat current J Q along the z(x)-axis, and the transverse temperature gradient along the x(z)-axis. In both cases, the magnetic field B is along the y-axis. f The temperature dependence of the thermal Hall angle for two different configurations at 12 T.

Inter-branch diffusion and anisotropic charge distribution
Even in the case of BP, 'extrinsic' scenarios cannot be excluded. Since the inversion center is not at an atomic site, a vacancy breaks the local inversion center and can generate skew scattering or side jump. In the scenario put forward by Guo et al. 25 , resonant coupling between phonons and dynamical effects, generates a 'side jump' thermal Hall effect, where the thermal Hall angle does not scale with longitudinal thermal conductivity, in agreement with experiments. However, the suppression of THE by the introduction of extrinsic atoms in strontium titanate 15,16 and the absence of correlation between the thermal Hall angle and the phonon mean-free-path constitute serious challenges for any 'extrinsic' scenario.
Let us now consider those features of BP, which can nourish an 'intrinsic' scenario. The phonon spectrum of BP 34 with a focus on energies below 100 cm −1 is shown in Fig. 4a. At 30 K, only the three acoustic modes (one longitudinal and two transverse) are thermally populated. Figure 4b shows the angle dependence of their wave-vector with an energy of k B T peak ≈ 20 cm −1 , which roughly corresponds to the peak temperature. As seen in the figure, two acoustic modes, the longitudinal (LA) and a transverse mode (TA1) display a pronounced anti-crossing and become almost degenerate along the x-axis.
A second relevant feature is the spatial distribution of charge density in bulk BP. Computed and highlighted by Hu et al. 35 (see Fig. 4c), it is highly orientational. We also note that dipole-active phonon modes have been observed in BP by infrared spectroscopy 36 , suggesting the presence of unevenly distributed positive and negative charges. Therefore, even in this covalent solid, the magnetic field can couple to phonons through its influence on charge distribution. Interestingly, Valagiannopoulos et al. 37 have found that there is a remarkable anisotropy of electromagnetic propagation in black P. They calculated the real and imaginary parts of the dielectric constant and found that an electromagnetic wave is damped much less along the 'zigzag' direction than along the 'armchair' direction (see the Supplementary Material for more details).
Note that at peak temperature, the phonon mean-free-path is well below the sample dimensions (see the Supplementary Material for more details), implying a source of thermal resistivity in addition to boundary scattering. Inter-branch coupling between harmonic vibrations was recently invoked to explain the glass-like thermal conductivity of many crystalline solids 26,27 . Our result calls for a theoretical examination of the role of magnetic field in such a context.

Discussion
In magnetic insulators, collective excitations may couple to phonons and generate a thermal Hall effect 24 . In contrast, in BP, as well as in strontium titanate, phonons are the only identified collective excitations. The two solids share at least two uncommon features. In both, the phonon mean-free-path is not a monotonous function of temperature, which has been tracked to abundance of momentumconserving phonon-phonon collisions 34,38 . In both, there is a non-trivial coupling between distinct phonon modes 39 .
We saw that the transverse and longitudinal conductivities peak at the same temperature in a variety of insulators. This strongly suggests a prominent role played by heat-carrying acoustic phonon in generating the THE even in magnetic materials. However, this does not exclude a role played by magnetism. In magnetic materials, a phonon can couple to another bosonic mode (and possibly a magnon) specific to each material.
In summary, phonons of black phosphorus generate a κ ij larger than what was reported in any other insulator. The ratio of κ ij to κ jj in this system is comparable to other insulators and in all cases, the longitudinal and transverse thermal conductivities peak at the same temperature where the phonon wavelength and the magnetic length are comparable in size. The result shortens list of ingredients required to produce a phonon THE. With warming towards room temperature a difference arises, presumably due to non-negligible contribution of the thermoelectric response to transverse heat flow. b Comparison of the temperature dependence of κ xz , multiplied by −150 and κ xx . c Comparison of the temperature dependence of κ zx , multiplied by 1000 and κ zz .

Measurements
All transport experiments were performed in a commercial measurement system (Quantum Design PPMS) within a stable high-vacuum sample chamber. Electrical transport responses were measured by a standard four-probe method using a current source (Keithley6221) and a DC-nanovoltmeter (Keithley2182A). In thermal transport measurements, both one-heater-three-thermocouples (type E) and oneheater-three-thermometers (Cernox 1030) techniques were employed to simultaneously measure the longitudinal and transverse thermal gradient. The thermal gradient in the sample was produced through a 4.7 kΩ chip resistor alimented by a current source (Keithley6221). The DC voltage on the heater and thermocouples (thermometers) was measured through the DC-nanovoltmeter (Keithley2182A). The thermocouples, the heat-sink, and the heater were connected to samples directly or by gold wires with a 50 microns diameter. All contacts on the sample were made using silver paste. Error bars in the main figures represent one standard deviation.
The longitudinal ( ∇ T i = (T 3 − T 2 )/l) and the transverse ( ∇ T j = (T 1 − T 2 )/w) thermal gradient generated by a longitudinal thermal current J Q were measured. They lead to the longitudinal (κ ii ) and the transverse (κ ij ) thermal conductivity, as well as the thermal Hall angle ( ∇ T j / ∇ T i ): Here l, w, Q are the distance between longitudinal thermocouples, the sample width and the heat power respectively.  . In each case, κ ij (T) is multiplied by an amplification factor a varying from (−)300 to (−)1000.

Calculations
The phonon dispersions were obtained using the dynamical matrices calculated in ref. 34 . These calculations were performed using density functional perturbation theory 40 as implemented in the QUANTUM ESPRESSO package 41 . The Perdew, Burke, and Ernzerhof's generalized gradient approximation 42 and Garrity et al.'s pseudopotentials 43 were used. The van der Waals interaction was taken into account using Grimme's semiempirical recipe 44 . Planewave cutoffs of 50 and 250 Ry were used for the basis-set and charge density expansions, respectively. A 12 × 12 × 12k-point grid was used for the Brillouin zone integration in the self-consistent density functional theory calculations with a Marzari-Vanderbilt smearing of 0.02 Ry. The dynamical matrices were calculated on an 8 × 8 × 8q-point grid, and the phonon dispersions and density of states were obtained by Fourier interpolation.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. The three acoustic branches are plotted with different colors. This spectrum, first calculated in ref. 34 is in agreement with a recent X-ray experimental study 48 . b Angle dependence of the phonon wave-vectors with a frequency of 20 cm −1 projected in the xz plane. The longitudinal mode (in black) and a transverse mode (in red) are almost degenerate along x. Energy transfer across phonon branches can be affected by the magnetic field. c Spatial distribution of charge concentration in bulk BP 35 . Note the x-z anisotropy. The (c) is adapted from ref. 35 , Royal Society of Chemistry.